3.7.60 \(\int \frac {x^7}{(a+c x^4)^2} \, dx\) [660]

Optimal. Leaf size=33 \[ \frac {a}{4 c^2 \left (a+c x^4\right )}+\frac {\log \left (a+c x^4\right )}{4 c^2} \]

[Out]

1/4*a/c^2/(c*x^4+a)+1/4*ln(c*x^4+a)/c^2

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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} \frac {a}{4 c^2 \left (a+c x^4\right )}+\frac {\log \left (a+c x^4\right )}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + c*x^4)^2,x]

[Out]

a/(4*c^2*(a + c*x^4)) + Log[a + c*x^4]/(4*c^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^7}{\left (a+c x^4\right )^2} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{(a+c x)^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-\frac {a}{c (a+c x)^2}+\frac {1}{c (a+c x)}\right ) \, dx,x,x^4\right )\\ &=\frac {a}{4 c^2 \left (a+c x^4\right )}+\frac {\log \left (a+c x^4\right )}{4 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.82 \begin {gather*} \frac {\frac {a}{a+c x^4}+\log \left (a+c x^4\right )}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + c*x^4)^2,x]

[Out]

(a/(a + c*x^4) + Log[a + c*x^4])/(4*c^2)

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Maple [A]
time = 0.14, size = 30, normalized size = 0.91

method	result	size
default	\(\frac {a}{4 c^{2} \left (x^{4} c +a \right )}+\frac {\ln \left (x^{4} c +a \right )}{4 c^{2}}\)	\(30\)
norman	\(\frac {a}{4 c^{2} \left (x^{4} c +a \right )}+\frac {\ln \left (x^{4} c +a \right )}{4 c^{2}}\)	\(30\)
risch	\(\frac {a}{4 c^{2} \left (x^{4} c +a \right )}+\frac {\ln \left (x^{4} c +a \right )}{4 c^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*a/c^2/(c*x^4+a)+1/4*ln(c*x^4+a)/c^2

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Maxima [A]
time = 0.29, size = 32, normalized size = 0.97 \begin {gather*} \frac {a}{4 \, {\left (c^{3} x^{4} + a c^{2}\right )}} + \frac {\log \left (c x^{4} + a\right )}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

1/4*a/(c^3*x^4 + a*c^2) + 1/4*log(c*x^4 + a)/c^2

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Fricas [A]
time = 0.38, size = 35, normalized size = 1.06 \begin {gather*} \frac {{\left (c x^{4} + a\right )} \log \left (c x^{4} + a\right ) + a}{4 \, {\left (c^{3} x^{4} + a c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

1/4*((c*x^4 + a)*log(c*x^4 + a) + a)/(c^3*x^4 + a*c^2)

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Sympy [A]
time = 0.13, size = 29, normalized size = 0.88 \begin {gather*} \frac {a}{4 a c^{2} + 4 c^{3} x^{4}} + \frac {\log {\left (a + c x^{4} \right )}}{4 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(c*x**4+a)**2,x)

[Out]

a/(4*a*c**2 + 4*c**3*x**4) + log(a + c*x**4)/(4*c**2)

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Giac [A]
time = 0.89, size = 48, normalized size = 1.45 \begin {gather*} -\frac {\frac {\log \left (\frac {{\left | c x^{4} + a \right |}}{{\left (c x^{4} + a\right )}^{2} {\left | c \right |}}\right )}{c} - \frac {a}{{\left (c x^{4} + a\right )} c}}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a)^2,x, algorithm="giac")

[Out]

-1/4*(log(abs(c*x^4 + a)/((c*x^4 + a)^2*abs(c)))/c - a/((c*x^4 + a)*c))/c

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Mupad [B]
time = 0.05, size = 29, normalized size = 0.88 \begin {gather*} \frac {\ln \left (c\,x^4+a\right )}{4\,c^2}+\frac {a}{4\,c^2\,\left (c\,x^4+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + c*x^4)^2,x)

[Out]

log(a + c*x^4)/(4*c^2) + a/(4*c^2*(a + c*x^4))

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